Parity Games of Bounded Tree-Depth

Abstract

The exact complexity of solving parity games is a major open problem. Several authors have searched for efficient algorithms over specific classes of graphs. In particular, Obdržálek showed that for graphs of bounded tree-width or clique-width, the problem is in $\mathrm{P}$, which was later improved by Ganardi, who showed that it is even in $\mathrm{LOGCFL}$ (with an additional assumption for clique-width case). Here we extend this line of research by showing that for graphs of bounded tree-depth the problem of solving parity games is in logspace uniform ${\text{AC}}^0$. We achieve this by first considering a parameter that we obtain from a modification of clique-width, which we call shallow clique-width. We subsequently provide a suitable reduction.

This is the full version of the paper that has been accepted at CSL 2023 and is going to be published in Leibniz International Proceedings in Informatics (LIPIcs)